Problem 100 - Algebra for College Students

Question

Solve each equation by locating the x-intercepts on a calculator graph. Round approximate answers to two decimal places. $$x^{2}-3 x^{1 / 2}-12=0$$

Step-by-Step Solution

Answer

The approximate x-intercepts are at x = -2.43 and x = 5.43.

Step 1: Graph the Equation

Enter the equation \( y = x^{2} - 3x^{1/2} - 12 \) into a graphing calculator and plot the graph.

Step 2: Find the x-intercepts

Locate the points where the graph crosses the x-axis. These points are the x-intercepts and represent the solutions to the equation.

Step 3: Approximate the x-intercepts

Use the calculator's 'zero' or 'root' function to find the exact x-values of the x-intercepts. Round these values to two decimal places.

Key Concepts

x-intercepts

When solving quadratic equations, one vital concept is the x-intercepts. X-intercepts are the points where the graph of the equation crosses the x-axis.
These intersections represent the values of 'x' that make the equation true, meaning they are the solutions to the equation.
In other words, they are the roots of the equation.

To find the x-intercepts, we set the equation to zero and solve for 'x'.
For the given equation, when we graph it, the points where the curve touches the x-axis are our x-intercepts.

In our example:

• Graph the equation: Plot the function on the graphing calculator.
• Locate x-intercepts: Identify where the curve crosses the x-axis.
• Use the calculator's functionality: Use the calculator's 'zero' or 'root' function to pinpoint the exact values.

Noting these points helps us understand the solutions to the quadratic equation.

graphing calculator

A graphing calculator is a powerful tool in solving quadratic equations. It makes the process visual and tangible by providing a graph of the equation.

Here’s how to use it effectively in our example:

• Input the equation: Enter the equation in the form y = f(x).
• Adjust the window: Make sure your viewing window shows all relevant parts of the graph where it may intersect the x-axis.
• Plot the graph: Use the graphing feature to visualize the equation.
• Identify intersections: Look for the points where the graph crosses the x-axis, as these are the x-intercepts.

The graphing calculator not only finds solutions but also helps you understand the shape and behavior of the equation's graph.

approximation

In mathematics, especially when dealing with graphs, approximation plays a crucial role. It helps when exact values are complex or impossible to determine with elementary methods.

For our quadratic equation, finding x-intercepts may not result in clean, easy numbers.
That's where we use approximations:

• Locate the intercepts: On the graph, identify the x-intercepts visually.
• Use calculator functions: The 'zero' or 'root' features on graphing calculators provide numerical values for these intersections.
• Round the values: Since exact solutions might be irrational, we round the approximations to a given degree of accuracy, typically to two decimal places.

By approximating, we get practical solutions that are very close to the true values. In many real-world applications, these approximations are sufficiently accurate for making decisions or further calculations.